Halphen pencils on quartic threefolds.pdfVIP

a r X i v : 0 8 0 2 .1 0 4 9 v 2 [ m a t h .A G ] 7 F e b 2 0 0 8 HALPHEN PENCILS ON QUARTIC THREEFOLDS IVAN CHELTSOV AND ILYA KARZHEMANOV Abstract. For every smooth quartic threefold, we classify all pencils on it whose general element is an irreducible surface birational to a smooth surface of Kodaira dimension zero. 1. Introduction Let C be a smooth curve1 in P2 that is defined by a cubic homogeneous equation f(x, y, z) = 0, and let P1, · · · , P9 be nine distinct points on the curve C such that the divisor 9∑ i=1 Pi ?OP2 ( 3 )∣∣∣ C is a torsion divisor of order m 1 on the curve C. Then there is a curve Z ? P2 of degree 3m such that multPi(Z) = m for each point Pi. Let P be the pencil given by the equation λfm ( x, y, z ) + μg ( x, y, z ) = 0 ? Proj ( C[x, y, z] ) ～= P2, where g(x, y, z) = 0 is a homogeneous equation of the curve Z and (λ, μ) ∈ P1. Then a general curve in P is birational to an elliptic curve. Remark 1.1. The construction of pencil P can be generalized to the case when C has at most ordinary double points and the points P1, · · · , P9 are not necessarily distinct (see [3]). The pencil P is called a standard Halphen pencil. Definition 1.2. A Halphen pencil is a one-dimensional linear system whose general element is an irreducible subvariety that is birational to a smooth variety of Kodaira dimension zero. The following result is proved in [3]. Theorem 1.3. Every Halphen pencil on P2 is birational to a standard Halphen pencil. Let X be a smooth quartic threefold in P4. Then X is not rational, because Bir ( X ) = Aut ( X ) due to [5]. The following threefold analogue of Theorem 1.3 is proved in [1]. Theorem 1.4. Suppose that X is general. Then every Halphen pencil on X is cut out by λf1 ( x, y, z, t, w ) + μg1 ( x, y, z, t, w ) = 0 ? Proj ( C[x, y, z, t, w] ) ～= P4, where f1(x, y, z, t, w) and g1(x, y, z, t, w) are linearly independent linear forms, and (λ, μ) ∈ P 1. The assertion of Theorem 1.4 may fail without the generality assumption (se